Question

It is assumed that a constant density planetary body of radius \(a\) has a core of radius \(b\). There is uniform heat production in the core but no heat production outside the core. Determine the temperature at the center of the body in terms of \(a, b, k, T_{0}\) (the surface temperature), and \(q_{0}\) (the surface heat flow).

Short answer

Integrate heat equations, apply boundary conditions, and solve for constants to find center temperature in terms of given variables.

Step-by-step solution

Understand the Problem

We have a planet with a core and a mantle, where heat is produced uniformly in the core and no heat is produced outside the core. We need to find the temperature at the center of the planet, using the given parameters.

Identify Given Variables

We have the following variables: - Radius of the body: \(a\)- Radius of the core: \(b\)- Thermal conductivity: \(k\)- Surface temperature: \(T_0\)- Surface heat flow: \(q_0\)

Apply Heat Equation in the Core

In the core, the heat conduction equation with constant heat production per unit volume \(q_0\) is: \[-k \frac{dT}{dr} = q_0 \cdot r + C_1\]where \(r\) is the radial coordinate and \(C_1\) is an integration constant.

Integrate and Find Temperature Distribution in Core

Integrate the equation:\[\int -k \frac{dT}{dr} \, dr = \int (q_0 \cdot r + C_1) \, dr\] This gives:\[kT = \frac{q_0}{2} r^2 + C_1r + C_2\]where \(C_2\) is another integration constant.

Apply Heat Equation Outside the Core

Outside the core, there is no heat production, so the heat conduction equation is:\[-k \frac{dT}{dr} = C_3\]Integrating gives:\[kT = C_3r + C_4\]

Apply Boundary Conditions

1. At the core-mantle boundary \(r = b\), temperature from the core and outside should be equal:\[\frac{q_0}{2} b^2 + C_1b + C_2 = C_3b + C_4\]2. At the surface \(r = a\), temperature should equal the surface temperature \(T_0\):\[kT_0 = C_3a + C_4\]

Solve for Constants

Using the boundary conditions, solve the simultaneous equations to find values of integration constants \(C_1, C_2, C_3, C_4\). This step involves substituting back and forth between equations obtained in step 6.

Calculate Temperature at the Center

Substitute \(r = 0\) into the temperature equation for the core to find the temperature at the center:\[T_{center} = \frac{q_0}{2k} \cdot 0^2 + C_1 \cdot 0 + C_2\]The final expression simplifies to be based on \(b, q_0, a, k, T_0\) after constants are found.

Key concepts

Thermal Conductivity

Thermal conductivity, denoted as \(k\), is a crucial property in understanding heat flow in planetary bodies. It refers to how well a material can conduct heat. In the context of a planet, thermal conductivity affects how heat moves from its core to its surface. A higher rate means heat is transferred quickly, which might affect the surface temperature and overall temperature distribution.

In our planetary example, the core generates heat, and the heat must pass through the mantle to reach the surface. The mantle's thermal conductivity will dictate the rate at which this process occurs. A strong grasp of thermal conductivity helps to predict the temperature and heat flow within a planet's layers. This knowledge is essential for finding changes in temperature as you move from the core to the surface.

Understanding this concept allows us to frame and solve heat conduction equations like the ones in our exercise. It's about knowing how effectively a planet can move its heat internally.

Core-Mantle Boundary

The core-mantle boundary is the dividing line between the core and the mantle of a planet. It is significant because the core and the mantle have different thermal properties, mainly due to distinct heat production activities.

In our scenario, heat is produced only in the core and must travel to the mantle before reaching the surface. At the core-mantle boundary, these two regions must match temperature conditions, forming a critical boundary condition for our calculations. This involves ensuring that the temperature computed within the core matches the temperature initiated as it moves into the mantle.

• This boundary is where the two heat equations from the core and the mantle meet and must be solved simultaneously to find the correct temperature profile.
• Here, the continuity of temperature is used to solve for the integration constants in the heat equations, which ultimately determine the temperature at the center of the planet.

Understanding this boundary's role is vital for predicting the planetary heating and cooling processes.

Surface Temperature

Surface temperature \(T_0\) is a fixed condition and acts as a reference point in planetary heat conduction problems. It determines the heat exchange between the planet and its surroundings. By using \(T_0\), one can derive how the internal heat generated in the core affects the planet’s outermost layer.

In our exercise, the temperature at the surface is an initial condition used in solving for the temperature distribution throughout the planet. This parameter helps set up boundary conditions at the planet's surface so that we can calculate the planetary temperature profile effectively.

• Surface temperature is measured at \(r = a\), where \(a\) is the planet’s radius.
• It ensures that the mathematical solution aligns with physical reality, denoting how planetary heat reaches a thermodynamic equilibrium with the environment.

Typically, surface temperature impacts planetary atmospheres and climates, making it an essential factor for understanding global thermal dynamics.

Heat Production

Heat production in planets generally occurs due to radioactive decay or residual heat from the planet's formation. It is crucial because it's the primary source of thermal energy within the planetary core. In this exercise, we assume a uniform heat production within the core and no heat production outside it.

The heat produced generates a thermal gradient that drives the heat conduction towards the surface. In our model, this is represented by the surface heat flow \(q_0\) and influences the temperature calculation.

• Uniform heat production leads to uniform distribution and a simple cubic relation in the temperature equation within the core.
• The rate of heat production impacts the integration constants and therefore the core's temperature distribution.

This understanding is necessary to construct the equation for heat conduction that predicts how efficiently the planet cools down or heats up internally.

Integration Constants

In solving differential equations, like those used for planetary heat conduction, integration constants come into play after we integrate the equations. They are crucial for pinpointing exact temperature values in the planet's heat distribution model.

To find these constants, boundary conditions are applied, like the core-mantle boundary and surface conditions in our exercise. As we know:

• \(C_1, C_2\) related to the core, and \(C_3, C_4\) representing conditions in the mantle need precise values.
• The boundary conditions stemming from the surface temperature \(T_0\) and heat flow \(q_0\) help solve these constants, giving meaning to our equations.

Once calculated, they allow us to determine the temperature at any point within the planet, including the core center. Their accuracy is vital for predicting dynamic thermal processes in planetary structures.